Hey Anna, I got this from the Merriam-Webster's on-line dictionary. It means to be 'extremely angry or irritated'.
Main Entry: fit
Function: adjective
Inflected Form(s): fit·ter; fit·test
Etymology: Middle English; akin to Middle English fitten
Date: 14th century
1 a (1) : adapted to an end or design : suitable by nature or by art (2) : adapted to the environment so as to be capable of surviving b : acceptable from a particular viewpoint (as of competence or morality) : PROPER <a movie fit for the whole family>
2 a : put into a suitable state : made ready <get the house fit for company> b : being in such a state as to be or seem ready to do or suffer something <fair fit to cry I was -- Bryan MacMahon> <laughing fit to burst>
3 : sound physically and mentally : HEALTHY
- fit·ly adverb
- fit to be tied : extremely angry or irritated
- fit to kill : in a striking manner <dressed fit to kill>

I don’t have any real etymological info, but here’s my guess as to its origin. “Fit to be tied” means angry, beside one’s self, and in a bad state which could lead to being out of control and “fit to be restrained” as in “tied up” or "fit to be tied."
__________________

Lea, Progress has been made since 2002, for now I have found the expression in two references, which don’t say a heck of a lot more than I did, but are more respectable than I am.

FIT TO BE TIED: Extremely angry, enough so to suggest that physical restraint might be indicated to prevent major damage. This expression originated in the late 19th century. James Joyce used it in ‘Ulysses (1922), “I was fit to be tied,” one of the more understandable expressions of feeling in that difficult book. (Facts in File Dictionary of Clichés)

FIT TO BE TIED: Incensed, enraged, livid, irate, very angry. This expression probably comes form the hospital practice of restraining patients who pose a danger to themselves or others. In its contemporary hyperbolic usage, ‘fit to be tied’ refers to anyone (not just a patient) who is extremely angry or who is acting irrationally, implying that if this person were in a hospital, he would be tied down for his own protection as well as for the protection of others. “It threw the boss into a tizzy. . . . The boss is fit to be tied. When he gets hold of you. . . .” —C. Simak, ‘Strangers in the Universe,’ 1956 (Picturesque Expressions by Urdang)
_____________________